Macromolecules 1988,21, 1051-1059
-
already in the high-temperature region. Consequently, for
comparison of the rates of b* a* in polymers and monomer
models, R value should be compared at a much lower temperature where both polymers and monomer models are in the
low-temperature region. From the discussion above, the differences in R values between polymers and monomer models
should be larger at low temperature. Details of temperature
dependence will be published elsewhere.
(14) While the decay of the b* band is multiexponential and the
exact kinetic solution is difficult to obtain, the decay of the a*
1051
band is nearly constant at -5 ns for both Id and poly(MMAco-2) in EAc at 25 “C. Preliminary study on the temperature
dependence of R value reveals that back-reaction from the b*
state is negligible in these polymers at room temperature.
(15) Al-Hassan, K. A.; El-Bayoumi, M. A. Chem. Phys. Lett. 1980,
76, 121.
(16) Rettig, W. J.Lumin. 1980,26, 21. Rettig, W.; Marschner, F.
Nouv. J. Chim. 1983, 7,415.
(17) Rettig, W.; Rotkiewicz, K.; Rubaszewska, W. Spectrochim.
Acta, Part A 1984,40A, 241.
Adsorption of Block Copolymers in Selective Solvents
C. Marques’ and J. F. Joanny*t
Dgpartement de Physique des Matgriaux, Universitg Claude Bernard, 69622 Villeurbanne
Cgdex, France
L. Leibler
E.S.P.C.I., 10 rue Vauquelin 75231 Paris Cddex 05, France. Received August 7, 1987
ABSTRACT: We study the adsorption of A-B diblock copolymers on a solid plane in a highly selective solvent.
The A part is in a poor solvent and forms a molten layer on the solid wall where the solvent does not penetrate.
T h e B part, in a good solvent, forms a brush grafted on this molten layer. The structure of the adsorbed
copolymer film is governed by the chemical potential in the solution in contact with the wall. Copolymer
chains have a tendency to self-aggregate in this solution forming several different mesophases. We present
first a scaling theory of micelle and lamella formation and then study the geometry of the adsorbed film in
equilibrium with these two phases. An important issue of this work is the role of the van der Waals interaction
between the wall and the adsorbed A layer. If the copolymer asymmetry is large enough, Le., if the B part
has a higher molecular mass than the A part, in a wide range of concentration, and this quite independent
of the phase behavior of the solution, the thickness of the molten A layer results from a balance between the
van der Waals energy and the stretching energy of the brush. The B chains are almost fully extended in the
brush. Various other adsorption regimes are found when the asymmetry of the copolymer is decreased.
I. Introduction
Block copolymers provide a simple and versatile method
for steric stabilization of colloidal particles suspended in
a solution.l+? The most direct scheme for colloid protection
amounts to adsorbing a diblock chain from a solution in
a selective solvent. Then the insoluble block A (which we
sometimes call the “anchor”) precipitates near the particles
surface and adsorbs on it forming a dense film, whereas
the soluble block B (called the “buoy”) extends into the
solution and forms an external layer (“brush”). For example,3 for particles suspended in water, A may be an
aliphatic chain, while B may be a water-soluble polymer
such as POE or polyacrylamide. The relevant parameter
for steric stabilization is then the density of chains in a
“brush” u or equivalently the area per copolymer junction
2. For small 2 (i.e., Z < NB6f6a2
where NBis the “buoy”
polymerization index and a the monomer length) the
chains in the brush are stretched and when two protected
surfaces are brought together they repel1 each other. Such
interactions have been studied by using Israelachvili force
measurement apparatus for model block copolymers,
polystyrene-polyvinylpyridine in cyclohexane, by Hadzioannou et aL4 The aim of the present work is to estimate
2 , assuming that the protective layer is obtained by
equilibrium adsorption from a solution of diblock copolymers in a selective solvent. In particular, we study the
role of the molecular parameters of the chains, such as
their composition or molecular weight, and of the solution
concentration on the structure of the protective layer.
‘Present address: ENSL, 46, All6e d’Italie, 69364 Lyon Cedex 07,
France.
0024-9297/88/2221-1051$01.50/0
The structure of a protective layer in equilibrium with
a reservoir containing copolymer chains is determined not
only by the free energy of a chain in the layer but also by
the external chemical potentials of the chains and the
solvent. Hence adsorption processes are complicated by
the tendency of copolymer chains to self-associateand form
organized structures (micelles, lamellas, ...).5,6 For a description of the adsorption we thus need a model for both
the protective layer and the solution of copolymer chains.
We shall simplify the description by considering the case
of a highly selective solvent, very poor for A and very good
for B. Then the solvent does not penetrate the A layer
precipitated against the solid surface, which is essentially
a molten A film. We shall also neglect the concentration
gradients in the copolymer layer. An important issue of
this work is the role of long-range forces.’ We will consider
the simple case of van der Waals forces and neutral
polymers, but the results could easily be extended to
different kinds of long-range forces. In such a highly selective solvent, we have constructed a simple model of
self-association of the copolymer solution. We consider
two simple limiting cases, that of spherical micelles and
that of lamellar structures; we limit our considerations to
dilute solutions and situations close to the critical micelle
concentration.
The paper is organized as follows: In the next section,
we study the spreading of a copolymer droplet of finite
volume on the wall; in section 111, we derive the general
properties of the adsorbed copolymer layer when the
chemical potentials in the solution are known. These
chemical potentials are calculated in section IV for micellar
phases and lamellar phases and adsorption in equilibrium
0 1988 American Chemical Society
1052 Marques et al.
Macromolecules, Vol. 21, No. 4 , 1988
with these phases is discussed. Section V discusses the
effect of the penetration of the solvent in the molten anchor layer.
11. Structure of A Block Copolymer Wetting
Film: “Pancakes”
When the adsorbing wall is in contact with a pure solvent, a small droplet of block copolymer of finite volume
V is deposited on the wall. A t equilibrium, the drop
spreads and forms a very thin wetting film on the wall. We
study here the properties of this wetting film assuming that
no polymer is dissolved in the bulk of the liquid. Although
this is not a geometry where experiments are likely to be
made, we consider it here for two reasons:
i. The absence of copolymer in the solvent makes the
problem quite simple and one can compare the relative
importance of the interaction of the copolymer with the
solid surface and of the entropic contribution of the copolymer to the free energy.
ii. The equilibrium thickness of the film in this geometry plays a major role in the general discussion of the
statility of block copolymer adsorbed layers which will be
made later on.
As already mentioned the diblock copolymer is composed of two chemically different chains A and B. The
B part has a polymerization index N B and is in a good
athermal solvent; when the copolymer is free in the solvent,
the radius of gyration of the B part is given in a Flory
approximationby Rm = NB3I5a,
a being a monomer length.
The B monomers are repelled by the wall. The A part has
a polymerization index NA and is totally incompatible with
the solvent. As it is often the case in practice, polymers
A and B are highly incompatible, one thus expects the
formation of regions of pure molten polymer A where
neither the solvent nor monomers B penetrate. This of
course is an extreme limit, the molten A layers are always
slightly swollen by the solvent, we will discuss this effect
in section V, it will be neglected here. The A monomers
are strongly attracted to the wall. The characteristic radius
of the A part of the polymer is R G A = NA112a.The asymmetry between the two parts of the copolymers is measured
by
0 = RFB/RGA
= Ng3l5/N~’l2
When @ = 0 (homopolymer A) the anchor forms on the
solid wall a thin film which has been called a pancake.8
The important physical parameters are the three interfacial tensions yWA,yws, and y M (between wall and polymer
A, wall and solvent, and polymer A and solvent) and more
precisely the spreading power S defined as S = yws - yAS
- YWA and the long-range interactions between the wall,
the polymer A, and the solvent. We study here the simple
case where these interactions can be described only in
terms of van der Waals forces by a negative Hamaker
constant AWAs = -A such that the energy of a film of
thickness e per unit surface is written as
P(e) = -AwAs/127re2 = A/127re2
(1)
The van der Waals force has thus a thickening effect on
the adsorbed layer. When the spreading power S is positive (the only case which will be considered throughout
the paper), the equilibrium thickness of the homopolymer
A pancake is
eo = (A/47rS)’i2
(2)
Notice that although this thickness eo is of the order of a
few monomer lengths a , i.e., much smaller than a chain
radius RGA, no effect related to the confinement of the
polymer has been included. de Gennesg has shown that
t
t
+
t
t
t
+
t
t
t
t
t
Figure 1. “Pancake formation”. The molten A pancake has a
thickness e; the grafted B brush has a thickness L. The surface
density is u = ( a / D ) 2 .
for molten polymers these effects exactly vanish. The
end-to-end distance in the direction parallel to the plane
is of the order of R G A .
For true copolymers 0 # 0 (we will restrict ourselves
from now on to situations where > 1))the A part of the
copolymer forms a thin pancake of thickness e where
neither the solvent nor the B monomers penetrate. The
B parts of the copolymer, the buoy, forms thus a grafted
layer of thickness L outside the pancake (see Figure 1).
We call D = Z1lz the average distance between “heads”
(junction points of the copolymer), the surface density is
u = a2/D2. The pancake is composed only of molten
polymer A so that
NAua/e = 1
(3)
The structure and energy of grafted layers with a surface
density u have been studied in detail by Alexanderro(see
Appendix A). When u < u1 NB+l5the B chains are
unperturbed. When u >> u1 the chains are stretched; the
layer equilibrium thickness is
-
N B ~ u ’ ~ ~
(4)
The energy per unit surface of the layer is (throughout the
paper, the temperature unit is such that the Boltzmann
constant k B is equal to 1)
T
FBs = -NBu11/6
(5)
a2
The total energy of the copolymer pancake may then be
written as
L
34 is the area covered by the pancake on the plane related
to the total volume of monomers A by VA = e&, the first
two terms account for the capillary energies (the second
term accounts for the presence of junction points and is
proportional to their surface density), the third term is the
van der Waals energy (we have neglected the contribution
of the grafted layer where the concentration is small), and
the last term is the energy of the grafted layer. There is
no entropic contribution of polymer A in this surface energy; as already explained, there is no confinement energy;
if the layer thickness e is larger than the radius of the A
chains, a stretching energy F A s should be included this
term is always negligible here.
The equilbrium thickness e is obtained by minimization
of the free energy (6) at constant volume VA:
s = -A- 47re2
6 a2
(7)
Block Copolymer Adsorption 1053
Macromolecules, Vol. 21, No. 4, 1988
We consider here only complete wetting situations S > 0
(monomers A are strongly attracted by the wall) and assume ANA/T >> 1which is usually true in real situations;
this leads to two different regimes.
a. HomopolymerRegime. When the spreading power
S is larger than a critical value
S, = -@20/23(ANA/T)11/23
T
&AZ
I
+
the effect of the buoy is small and the pancake thickness
is e = eo. The thickness of the grafted layer is
The chains are extended in the grafted layer, this of course
requires u > crl or S < (A/a2)B4otherwise the B chains
remain unperturbed.
b. van der Waals-Buoy Regime. When the spreading
power S is smaller than S,, it becomes negligible in eq 7,
the pancake thickness results from a balance between the
grafted layer stretching energy and the van der Waals
energy
e = a(ANA/T)6/23p10/23 NA11/23NB -6123
(9)
N
The thickness of the grafted layer is
L = NB21/23NA-4/23aN Rm@12/23(ANA/T)2/23(ga)
In this regime, the thichess of the grafted layer scales with
NB with an exponent slightly smaller than 1. The thickness of the molten A layer decreases with NB.
In practice if both parts of the copolymer are long
polymers and if the copolymer is not too dissymmetric (p
< NA3l5),S should be larger than S, and the pancake
thickness depends only on the anchor A. The other regime
is obtained for very dissymmetric copolymer @ >> NA3J5;
we will not consider such polymers in the following.
111. Block Copolymer Adsorption from Solution:
General Results
In most experiments, the adsorbing wall is in contact
with a bulk copolymer solution which plays the role of a
reservoir. The whole wall is covered by a copolymer layer;
the precise structure of the adsorbed layer depends on the
structure of the solution in the reservoir. At very low bulk
concentration, the solution is isotropic and chains are
isolated; it then reaches a critical micellar concentration
(cmc) located well in the dilute regime. A t higher concentrations the copolymer forms organized mesophases
with various structure^.^,^ In the present section, we
characterize the reservoir by the chemical potential of the
polymer chains pexand its osmotic pressure rex;
these two
quantities should then be related to the bulk concentration
in each region of the phase diagram. The phase diagram
of diblock copolymers being very rich and quite complicated, we do not consider here all possible mesophases, but
limit the study in the next section to very dilute solutions
both below and above the cmc and to the simplest possible
organized phase, a lamellar phase.
1. Structure of the Adsorbed Layer. In the presence
of a reservoir where the chemical potential is pexand the
osmotic pressure rex,
the equilibrium structure of the block
copolymer layer minimizes the grand canonical free energy"
9 = 3 - pexn+ r,,VB
(10)
3 is the total free energy of the copolymer layer, n the
number of adsorbed chains, and VBthe volume accessible
to the solvent. Assuming no penetration of the solvent into
d
A
+
+
+
+
+
+
+
+
+
Figure 2. Block copolymer adsorbed film. The anchor forms
a molten layer of thickness d on the surface; the B brush has a
thickness L.
the molten A layer, the volume VB is related to the
thickness of the grafted buoy layer by VB = L A (Figure
2). The area of the plane .A being constant, we consider
the grand canonical free energy per unit surface G = 9/34.
This is written as
G = -s - YAsu - peXua-'
+ P ( d ) + GBS(u,rex)+ FAs(d)
(11)
As in eq 6 , the first two terms represent the contribution
of the capillary energies and P ( d ) is the van der Waals
energy of the pure molten A layer of thickness d given by
eq 1 ( d is related to u by eq 3 ) .
FAs is the stretching energy12of the A chains which must
be included when d >> R A .
GBS = FBs + rexLis the Gibbs free energy of a grafted B
layer of surface density u under an external osmotic
pressure rex.It is studied in detail in Appendix A; if the
external solution is dilute, the external osmotic pressure
might be neglected, the thickness of the buoy layer is given
by eq 4, and the free energy GBs FBsis given by eq 5.
The equilibrium thickness of the anchor d or equivalently the equilibrium surface density u is obtained by
minimization of the free energy G keeping the chemical
potential per and the osmotic pressure rexconstant
-
(13)
According to the value of the effective chemical potential
p = (pex+
T , we distinguish four adsorption regimes
(Figure 3).
i. Rollin Regime. When p is negative and large p <
- pi
-(ANAIO 5 / 2 3 @ 3 0 / 2 3 the dominant term on the
right-hand side of eq 13 is the van der Waals energy. The
thickness of the A film is
d =u[AN~/T~]'/~
(14)
This is the same equation as the equation giving the
thickness of undersaturated helium films (first studied by
Rollin) on a solid substrate.13
ii. van der Waals-Buoy Regime. When the absolute
value of chemical potential p is small IpI < p1 = (ANA/
T)5/23p30/23,
the left-hand side becomes negligible in eq 13
and the equilibrium thickness d results from a balance
between van der Waals forces and the stretching energy
of the buoy FBs
d = (ANA/T)6/23p10/23a NB-6/23N
A1 1 / 2 3
(15)
N
iii. Buoy-Dominated Regime. When the chemical
potential p is positive and such that pL1< p < p2
Macromolecules, Vol. 21, No. 4, 1988
1054 Marques et al.
A n c h o r dominated
regime
i
\
4-
.-. . ......
~
A Core
Figure 4. Block copolymer micelle. The core of radius RM is
formed of molten polymer A; the external layer has a thickness
L.
thickness e. This determines a threshold for adsorption
in the chemical potential gexCsuch that
d(HexC) =
I
Figure 3. Variation of the effective chemical potential p as a
function of the anchor molten layer thickness d. Various adsorption regimes are found when ii is varied; no adsorption is
possible if d < e (p < gJ.
N A ~ the
/ ~dominant
~ ~ ~
term~ on~the, right-hand side of
eq 13 is the stretching energy of the buoy
d = p6/5p2a
(16)
iv. Anchor-Dominated Regime. At larger values of
p the stretching energy of the anchor dominates
d = NA1/2ap1/2
Of course, the maximum size of the A chains is when they
are fully stretched d = N,a, the anchor dominated regime
exists only if /3 << NA1/loor NB < NA.
2. Stability o f the Adsorbed Film. In any of these
adsorption regimes, the adsorbed copolymer film is stable
only if the potential G has a lower value at equilibrium
than in the absence of any adsorbed film where by construction G = 0.
In order to study this stability condition, we rewrite eq
11:
G = -S
+ g(d) - pexua-z
(W
where g(d) includes both the van der Waals energy and
the copolymer elastic energies.
The stability condition reads
dg
S > g(d) - d-
dd
It can now be directly checked from eq 7 that the pancake
thickness e calculated in section IIs is in this language such
that
dg
S = g(e) - ede
(One must of course neglect the small difference due to
the external pressure between FBsand GBSand include in
the pancake analysis the stretching energy of chains A
which was neglected in section 11.)
An adsorbed copolymer film on the wall is thus stable
if the equilibrium thickness d is larger than the pancake
e
As we have seen, e is generally in the range of a few angstroms, i.e., on the order of a few monomer lengths, and
the adsorption threshold will thus be in the Rollin regime.
This corresponds to extremely small concentrations as
shown below, it is thus probably not a measurable effect
for large NA where as soon as the block copolymer has a
finite concentration in the solution, it adsorbs on the wall.
IV. Block Copolymer Adsorption from Micellar
and Lamellar Solutions
The important parameter which governs the structure
of the adsorbed copolymer layer is the effective chemical
potential p = (pex + yAsa2)/T;we now calculate this
chemical potential for micellar solutions and lamellar
phases.
I. Adsorption from a Micellar Solution. a. Critical
Micelle Concentration. At extremely low bulk volume
fraction ab,the block copolymers do not aggregate and in
a first approximation different chains behave independently. The chemical potential of a chain is
pch
= T log
+ AF
(18)
AF is the free energy of one isolated chain. As we have
implicitely done earlier, we take as a reference state for
B chains isolated chains in a good solvent and for A chains
the molten state. The contribution of the B part to AF
is small, the contribution of the A part has been shown to
be dominated by surface proper tie^,'^,'^ in a very poor
solvent.
hF = ~ T ~ A ~ R A ~
In the extremely selective solvent which we consider, the
A part of the copolymer is collapsed and forms a small
molten globule of size R A = NA”3a so that
gch
= T log @b
+~
T ~ A ~ U ~ (184
N A ~ ~ ~
As the concentration is increased, it reaches the cmc
where micelles begin to form. We study the properties of
these micelles following a simplified theory quite similar
to the one proposed in ref 6. For the sake of simplicity,
we consider only a bimodal distribution of aggregates:
single chains and monodisperse micelles. Polydispersity
of the micelles could be treated along the lines of ref 24.
A given micelle is formed with p copolymer chains and its
core has a radius RM (Figure 4 ) . The junction points of
the copolymer are located on the surface of the core; the
surface density is u = p a 2 / 4 r R M 2= p113/NA213.
Block Copolymer Adsorption 1055
Macromolecules, Vol. 21, No. 4, 1988
The buoy B forms a grafted layer in a good solvent
around this spherical core. The properties of such grafted
layers were studied in detail by Daoud and Cotton;16 the
main results are rederived in Appendix B.
Whenever the concentration of the micelles is small, each
micelle might be considered independently as being in
equilibrium with a reservoir composed of the free chains
in solution. The chemical potential pex imposed by the
reservoir is related to the concentration of free chains by
eq 18. The number of chains in a micelle p minimizes the
grand canonical free energy
= F M - pe$
+ 4/33*[(R~+ L)3-
= ~'~T')'A~RM~
+ FBs
(24)
This leads to a surface density
Finally, the grafted layer thickness L is, by use of eq B-4,
(19)
where F M is the free energy of a micelle. The effect of the
reservoir osmotic pressure rexis small (see Appendix B)
and will be neglected hereafter.
With only leading order terms the micelle free energy
F M is written as
FM
The micelle core radius is
(20)
The first term is the interfacial energy between the molten
A core and solvent; FBsis the energy of the grafted buoy
layer. We consider here the case where the grafted layer
thickness L is larger than the core radius RM. The energy
FBsis then given by eq B.5
FBs = Tp3/2 log NB3/5/NA'/3p2/'5
Several small corrections were neglected in the micelle free
energy FM: The contribution of the junction points in the
surface energy gives a correction 43*7A&M2.
A careful consideration of the long-range van der Waals
forces leads to curvature corrections to the surface tension
17YAS
YAS(1 - 6/RM); the length 6 is in general of the
order of a monomer length a , i.e., much smaller than the
micelle radius RM. The junction point entropy T p log u
is negligibly small when the surface density u is small. The
elastic energy of the chains in the micelle core has been
studied in ref 25. It can be checked that this contribution
is always negligible here.
The equilibrium value of p is obtained by minimizing
the free energy Q at fixed chemical potential per. Micelles
start to form when the value of at the minimum is
negative. These two conditions determine the critical
micelle concentration.
-.
where pcmc is the chemical potential at the cmc
It is larger than the core radius RM if RFB > N A ~X ~ / ~ ~
a(yAsa2/T)4/25
which is always the case when P >> 1.
In an extremely dilute solution, chains are isolated and
the chemical potential in the solution pex is given by eq
18 which can be rewritten
Above the cmc, in a first approximation, the concentration
of free chains remains constant and the chemical potential
is ex = pcmc.
In order to be more quantitative, one should not consider
only one isolated micelle but should include in the theory
the translational free energy of the micelles and the interactions between micelles; this leads to a slow increase
of the chemical potential above the cmc. We neglect this
effect here and suppose that above the cmc the micelles
have the structure that we have calculated exactly at the
cmc.
b. Adsorption of Block Copolymers from Micellar
Solutions. We now combine these results with the general
results of section I1 to study the structure of the film
adsorbed on the wall from a micellar solution.
The chemical potential pCmcgiven by eq 22 is positive
and much larger than the thermal energy kT. As we have
focused on polymers such that N B > NA only two adsorption regimes are obtained, the buoy-dominated regime
= PI and the van der
if < NA~/~'(A/T)-'/~(YAS~~/T)~~/~'
Waals-buoy regime if P > P1.
In general we expect A / T 1and YAsa2/T 1. If P
is of order 1 the adsorbed layer is in the buoy dominated
regime, but as soon as P becomes much larger than 1van
der Waals forces become important.
i. Buoy-Dominated Regime. The thickness of the
anchor layer is from eq 16,
-
d = aNA'2/25P2(rAsa2/T)'8/25
-
(25)
P being close to 1, this thickness is approximately equal
to the Gaussian radius of the A chains R C A = NA1I2a.The
surface density u scales as
The number of copolymer molecules per micelle at the
cmc18 is
= NA-'3/25P2(rAsa2/T)'8/25
(26)
This is slightly smaller than the surface density in the
micelles given by eq 24. The area per chain I: = a-lu2 is
thus larger on the adsorbing wall than in the micelles.
The thickness L of the grafted layer is, from eq 3,
L = R F B N A ~ / ~ ~ ( ~ A ~ ~ (27)
~ / T ) ~ / ~ ~
If we neglect the logarithmic corrections, the value of the
cmc and the number of chains per micelle p depend only
on the molecular mass of the anchor NA. The volume
fraction at the cmc is exponentially small, as the cmc is
located well in the dilute regime.
which is the same thickness as that of the micelle grafted
layer (eq 24b). This has the same scaling behavior with
NB as the unperturbed Flory radius RFB (the chains are
not fully extended); however, this is much larger than the
Flory radius due to the NA dependence.
Macromolecules, Vol. 21, No. 4, 1988
1056 Marques et al.
ii. van der Waals-Buoy Regime. As soon as 0 >> 1,
van der Waals forces play a dominant role. The thickness
of the molten A layer is given by eq 9
d = a(ANA/T)6/23p10/23
Although this scales with NA1i2approximately as NA1/',
it is much smaller than the Gaussian radius due to the NB
dependence.
The surface density is
This again is smaller than the surface density in the micelle
given by eq 24. The thickness of the B layer is given by
eq 9a and is almost linear in NB ( N N B " / ~ ~ ) .
2. Block Copolymer Adsorption from a Lamellar
Phase. a. Structure of the Lamellas. Each lamella is
composed of a molten A layer of thickness 2dL and two
grafted B layers of thickness LL. It is in equilibrium with
the free chains in the solvent between different lamellae
where the chemical potential is per and the osmotic pressure rex.If we neglect any undulations of the lamellae,16
the thickness d L minimizes a grand canonical free energy
which has the same structure as eq 11
adsorbed layer than in the lamella.
V. Solvent Penetration in the Molten A Layers
Up to this point, we have considered that polymer A is
extremely incompatible both with polymer B and the
solvent and forms pure molten regions. Although this is
generally true for polymer B if NA and NB are large, there
is always a small penetration of the solvent in the molten
A region. The monomer concentration c in the anchor
layer on the adsorbing wall is then smaller than l/u3.
Equation 3 should be replaced by
NAu = dca2
(31)
If the chemical potential in the reservoir is pex and the
osmotic pressure is rex,
the grand canonical free energy
which gives the structure of the adsorbed layer is
+ P ( d ) + GBS(a,~,,)+ FAs +
G = - S - YAsa -
dJ'MIXA(d,C) + aexd (32)
The interfacial tensions S and YAS are slightly modified
by the dilution of the anchor layer. The Hamaker constant
A is also changed, it increases from 0 at zero concentration
to a positive value when ca3 = 1. FMmAis the mixing free
energy of the A monomers in the layer of thickness d. The
last term accounts for the penetration of the solvent in the
layer; the volume accessible to the solvent is now the total
volume V = A(L d). This free energy G depends on two
independent variables c and u.
Minimization with respect to c gives the balance of
pressures:
+
where uL = d L / u N Ais the surface fraction on each surface
of the lamella.
The main differences between the free energies of the
lamella and the copolymer on the adsorbed plane come
from the capillary energy and the van der Waals energy.
There is just one interfacial tension from the lamella
problem 7AS,
and the Hamaker constant AsAs is always
positive; the van der Waals energy has a thinning effect.
The equilibrium thickness of the lamellae is such that
The onset of lamellar phase formation would then require a comparison of the free energy of the lamellas with
that of any other possible phase. This is far beyond the
scope of this work. We will assume here that the lamellar
phase exists and has a given surface density uL. This might
not be the case for large asymmetries p >> 1. Equation 30
then gives the chemical potential in lamellar phase pexas
a function of aL. Notice, however, that lex
is in any case
larger than pcmc (the lamellar phase occurs always at a
higher concentration than the micellar phase) and that eq
30 imposes a minimum value for F~~
gT(
gASASNA
Pmin
=
220aT
5/23
)
p30/23
b. Block Copolymer Adsorption. The structure of
the copolymer adsorbed layer is obtained from eq 30 and
13. Two different regimes are found.
If pel is larger than p1 = T(ANA/T)5/23p30/23
(this will be
always the case if the two Hamaker constants are such that
A = -AwM << ASAS),the van der Waals terms are negligible
and dL = d ; the structure of the adsorbed polymer layer
is the same as the structure of the lamellar phase. This
result was already suggested by de Gennes.20
If pex < p l , pexis small and the adsorbed layer is in the
van der Waals-buoy regime described in the previous
section. The area per chain 2 = u2u-' is smaller in the
pA(c) is the osmotic pressure in the adsorbed layer
=
c2[a(FmA/c)ac],FAch= FA% is the elastic energy per chain
which depends only on d , and ad is the van der Waals
disjoining pressure i?d = -dP/dd.
As shown above the external pressure rexand the elastic
energy of the A chains are small; we will also assume that
the solvent is so selective that the disjoining pressure gives
a small contribution in the pressure balance. If this is not
the case, one must take into account the local variation
of the concentration c with the distance z to the wall and
in particular include gradient terms in the free energy.
Equation 33 might then be approximated by
rA(C) = 0
(34)
In the same poor solvent, a free A polymer would phase
separate into a dense polymer phase and almost a pure
solvent; eq 33 thus means that the concentration in the
adsorbed layer is that of this equilibrium dense phase.
More quantitatively,12 the osmotic pressure within a Flory
approach is given by a virial expansion
= 1/2uc2
+ 1/3w2c3
(35)
The second virial coefficient u is negative in a poor solvent;
it is often proportional to the shift from the 0 compensation temperature u
( T - 0)/0. The third virial
coefficient w u3 is positive. This leads to a concentration
KA(C)
-
-
c = -3v/2wz
Minimization of eq 32 with respect to
equilibrium thickness d
(36)
u
yields then the
(37)
This reduces to eq 13 when cu3 = 1. The only extra term
Block Copolymer Adsorption 1057
Macromolecules, Vol. 21, No. 4, 1988
being the chemical potential term pA(c) = dFMIXA/dc
calculated in the dense equilibrium phase, which simply
reflects the change in reference state for the A part of the
polymer which should not be chosen as the molten state
but as the equilibrium phase in the poor solvent.
The penetration of the solvent may be described as usual
in a poor solvent in terms of thermal blobsz1of size E
a4/lvl each containing g = ( l ~ l / a ~monomers
)-~
so that the
concentration c = g/E3 is given by eq 36.
All the results of section I11 can then be used if we
renormalize NA by the number of blobs NA/g and the
monomer length a by the blob size 5, both the Hamaker
constant A and the radius ratio @ being unchanged. (This
transforms eq 13 into eq 37.) The same renormalization
should be made for the determination of the external
potential pes in the study of micelle or lamella formation
in the reservoir. (The scaling theory predicts then y s =
TIE2.)
-
VI. Discussion
We have studied the equilibrium adsorption of diblock
copolymers in a highly selective solvent on a solid wall
taking into account possible mesophase formation in the
solution in equilibrium with the adsorbed polymer. Four
adsorption regimes have been found depending on the
value of the external chemical potential p,,.
In extremely dilute solutions the theory predicts a
threshold for adsorption which essentially depends on the
spreading power of the anchor A and on the strength of
the van der Waals energy of a film of polymer A in contact
with the wall. Just above this threshold, the adsorption
is controlled by the van der Waals energy (Rollin regime)
and the thickness of the anchor layer d varies as llog (Pbl-1/3,
being the bulk volume fraction (cf. eq 14 and 18b).
This, however, occurs at such a low concentration that it
is probably not observable in practice except for very short
anchor blocks.
At higher concentrations, chains self-aggregate to form
micelles in solution. We have built a simplified scaling
model of the critical micelle concentration predicting the
cmc the number of chains per micelle and the chemical
potential at the cmc as a function of the mass of the anchor
forming the core of the micelle. Two adsorption regimes
are found in equilibrium with a micellar solution depending on the asymmetry ratio @ of the copolymer (van
der Waals-Buoy regime and buoy-dominated regime).
When p is of order 1 the adsorption is governed by the
elastic energy of the buoy layer, the thickness of the molten
A layer is approximately equal to the radius of gyration
RGA(eq 25), and the thickness of the buoy layer scales as
the Flory radius of the B chains (eq 27). When @ >> 1 the
geometry of the adsorbed copolymer resulta from a balance
between van der Waals forces and the buoy layer elastic
energy; the thickness of the grafted layer increases approximately linearly with its molecular mass (eq 9 and 9a).
If the solution at a even higher concentration is in a
lamellar phase, the important result is that when van der
Waals forces are small the structure of the adsorbed layer
is the same as that of a lamella.
If the copolymer asymmetry is large enough, Le., if the
anchor has a much lower degree of polymerization than
the buoy, in a wide range of concentration in solution, the
adsorption occurs in the van der Waals-buoy regime; the
structure of the adsorbed layer is fairly universal in the
sense that it is independent of the concentration and copolymer organization in solution.
Quantitative comparisons could be made between the
theory presented here and experimental results, obtained
either by using a force measurement apparatus or colloidal
particles coated with copolymers. One of the main results
of our work is the dependence of the structure of the adsorbed film on the phase behavior of the copolymer solution, which is not very often considered. Another important point is the role of the long-range van der Waals
interactions. We considered here the case of thermodynamic equilibrium, which may be difficult to realize experimentally. These points certainly deserve further
theoretical study; experimentally a careful control of the
formation of the adsorbed layer is clearly needed.
Finally, we have considered only highly selective solvents
although we have discussed qualitatively a small swelling
of the molten layer by the solvent. In most experiments,
the solvent is not too selective; we wish to study the adsorption of block copolymers on nonselective solvents in
a future work.
Appendix A: Grafted Layers on a Flat Surface
We discuss here the structure and energy of a grafted
B layer (a brush) of known surface density u > u1 NB4l5,
under an external pressure rex.The external pressure
might be monitored either by osmotic effects or by an
external mechanical pressure as in a force measurement
apparatus.
The grand canonical free energy which should be minimized at equilibrium is
GS = FBS(L,u) a,,L
(A.1)
-
+
FBsbeing the free energy of the layer per unit surface.
We make here a blob description of the brush from the
work of Alexander.lo When the external pressure A,,
vanishes, each chain can be viewed as a fully extended
chain of excluded-volume blobs of size D = aa-1/2. Each
blob contains gD = (D/a)5/3 u-5/6monomers.
The thickness of the layer is
-
Each blob having an energy kBT, the free energy of the
layer per unit surface is
(A.3)
When the external pressure re,does not vanish, the blobs
are not deformed only if A,, C T/D3 and the effect of the
external pressure is negligible in (A.1). When a,, > T/D3,
the grafted layer forms a semidilute solution of concentration c = (NBU/LU2) where the osmotic pressure a =
T / ~ ~ ( c abalances
~ ) ~ / the
~ external pressure rex.This gives
the thickness of the adsorbed layer:
(A.4)
The free energy GBS scales as
GBS
- a,,L
[
TNBaa-2 A:3]5ig
(A.5)
This crosses over smoothly to the results at zero external
pressure when A,,
T/D3.
When we study copolymer adsorption in equilibrium
with a reservoir, the external pressure A,, is the osmotic
pressure of the dilute solution of free chains rex= Tcb/NB
which is always much smaller than T / D 3 ;the external
pressure a,, has thus always a small contribution which
can be neglected.
In a force measurement appratus one brings together
two grafted layers at a distance 2L. It can be shown2*that
Macromolecules, Vol. 21, No. 4, 1988
1058 Marques et al.
the two brushes do not interpenetrate. If the distance L
is smaller than the equilibrium distance L, given by eq
A.2 the force between the two layers is extremely small.
In order to obtain a distance L smaller than the equilibrium distance L, one must apply a finite pressure rex.
The
relation between the measured force and the distance is
given by eq A.4
(A.4a)
This holds for Lmh << L << Leq,Lminbeing the minimum
distance obtained when the chains are collapsed on the
surface: Lmin= NBaa.
When L becomes close to L-, the pressure rex
increases
strongly and cannot be described by equation A.4a. We
use a Flory model for the potential G
re&
log (1- ca3) + xc(1 - ca3)L + T
(A.6)
where x is the FLORY interaction parameter and the
concentration c is related to L by c = NB(T/La2. Minimization with respect to L gives a pressure which diverges
with distance as L approaches Lmin
T
a,, = - a3 log (1
-
%)
chains long enough that L >> Rw
The free energy density is F = kT/t3;the layer free
energy is
-4ar2 dr = Tp312log L
(B.5)
RM
The effect of a finite external pressure may be discussed
following the same procedure as in Appendix A. Blobs are
not deformed as soon as a,, < TIS3. This defines a
characteristic radius
If r < Re the structure of the grafted layer is given by eq
Bl-B3. If r > Re the concentration is roughly constant as
in the planar case
cea3 =
419
(
aexa37)
The conservation of monomers in the grafted layer leads
to a thickness L
(A.7)
and crosses over smoothly to (A.4a) when (L - L~,,)/Lmh
= 1.
Appendix B: Grafted Layer in a Spherical
Geometry
Grafted polymer layers on a sphere have been studied
in detail by Daoud et al.’6-23 The junction points are on
the core of the micelle with a surface density a; the radius
of the core is RM
In the absence of external pressure the grafted layer may
be described in terms of blobs quite similarly to the planar
case. The size of the blob, however, depends on the distance from the center of the micelle (Figure 4).
when the pressure rexis larger than a critical value a,
= NB-9/5p9/10- T
(B.7)
rex > a c
a3
This critical value acis much larger than the pressure at
the overlap concentration c*
NB4IS(p >> 1). The effect
of the external pressure might thus be neglected here, the
concentration of free chains being always smaller than c*.
-
Acknowledgment. We thank P.-G. de Gennes for
sharing with us some unpublished results and P.-G. de
Gennes, F. Brochard, D. Broseta, and M. Tirrell for stimulating discussions.
References and Notes
Each blob has excluded-volume statistics and contains g
monomers
The local concentration is
Each micelle contains p = a4aRM2a-?chains. The total
number of B monomers is thus
L,
RM+L
NBP
=
4ar2c(r)dr
This determines the thickness of the grafted layer L
In the last case there is no effect of the spherical geometry
.and the result is the same as for the planar geometry
(Appendix A). From now on, we restrict the study to B
Napper, D. Polymeric Stabilization of Colloidal Dispersion;
Academic: London, 1983.
Gast, A.; Leibler, L. Macromolecules 1986,19,686.
Vincent, B.; Clarke, J.; Barnett, K. Colloids Surf. 1986,17,51.
Hadzioannou, G.;Patel, S.; Granick, S.; Tirrell, M. J . Am.
Chem. SOC.1986,108,2869.
Leibler, L. Macromolecules 1980,13,1602.
Leibler, L.;Orland, H.; Wheeler, J. J. Chem. Phys. 1983,79,
3550.
Israelachvili, J. Intermolecular and Surface Forces; Academic:
London, 1985.
Joanny, J. F.; de Gennes, P.-G. CRAS 1984,299.11,279. de
Gennes, P.-G. Rev. Mod. Phys. 1985,57,827.
de Gennes, P.-G. CRAS 1980,290.B,509.
Alexander, S. J . Phys. (Les Ulis, Fr.) 1977,38,983.
Rowlinson, J. R.; Widom, B. Molecular Theory of Capillarity;
Oxford University Press: Oxford, 1982.
See, for instance: de Gennes, P.-G. Scaling Concepts in
Polymer Physics; Cornel1 University Press: Ithica, NY, 1978.
Brewer, D. F. In Physics of Liquid and Solid Helium; Benneman, K., Ketterson, J., Eds.; Wiley: New York, 1978.
Lifschitz, I. M.; Grosberg, A.; Khokhlov, A. Reu. Mod. Phys.
1978,50,685.
Joanny, J. F.; Brochard, F. J. Phys. (Les Ulis, Fr.) 42,1145.
Daoud. M.:Cotton. J. P. J. Phvs. (Les Ulis. Fr.) 1982.43,531.
This term ’is discussed in detail in ref 11.
This result was independently obtained by A. Halperin (preprint 1987).
Undulations of lamellas have been studied in detail for classical surfactants by Helfrich (Helfrich, W. J . Phys. (Les Ulis,
Macromolecules 1988,21, 1059-1062
Fr.) 1985,46, 156) and for copolymers by Cantor (Cantor, R.
Macromolecules 1981, 14, 1186).
(20) de Gennes, P.-G., unpublished results.
(21) de Gennes, P.-G. J. Phys. Lett. 1975,36, L.55; J. Phys. Lett.
1978, 49, L.299.
(22) de Gennes, P.-G.
Coll8ge de France lectures, 1984.
1059
(23) Pincus, P.; Witten, T . A. Macromolecules 1986, 19, 2609.
Pincus, P.; Witten, T . A.; Cates, M. Europhys. Lett. 1986,2,
137.
(24) Goldstein, R. J . Chem. Phys. 1986, 84, 3367.
(25) Semenov, A. N. Sou. Phys.-JETP (Engl. Transl.) 1985, 61,
733; Zh.Eksp. Teor. Fiz. 1985,88, 1242.
Adsorption of Polybutadienes with Polar Group Terminations on
the Solid Surface. 1. Infrared Study at the Silica Surface
Masami Kawaguchi,*t Minoru Kawarabayashi,t Nobuo Nagata,f
Tadaya Kate,+ Akira Yoshioka,f and Akira Takahashit
Department of Industrial Chemistry, Faculty of Engineering, Mie University,
1515 Kamihama-cho, Tsu, Mie 514, Japan, and Nippon Zeon Co., Ltd., Research and
Development Center, 1-2-1 Yako, Kawasaki-ku, Kanagawa 210, Japan.
Received M a y 22, 1987; Revised Manuscript Received October 21, 1987
ABSTRACT Adsorption of polybutadiene terminated with a very polar functional group (T-PBR)on a silica
surface from carbon tetrachloride solution has been investigated by using IR spectroscopy and compared with
normal polybutadienes (PBR). Carbon tetrachloride is a good solvent for polybutadiene. The adsorbed amount,
A , the surface coverage 0 of the silanol groups, and the bound fraction of a polymer chain p were measured.
A marked difference in adsorption behavior between T-PBR and PBR polymers was observed. T-PBR has
an A and 0 twice as large as those for PBR, while the p value was approximately the same for both polymers.
From the chemical structure of the terminal group we can qualitatively interpret the difference in the adsorption
behavior by taking into account the preferential adsorption of the terminal polar functional groups over the
double-bond groups in polybutadiene chains onto the silanol groups of silica surface.
Introduction
The loop-train-tail conformation is generally accepted
as a structure of flexible polymer chains adsorbed on flat,
cylindrical, and spherical surfaces if there is no specific
interaction between polymer and surface.l+ Tail portions
of adsorbed polymer chain are considered to be randomflight chains attached on a surface with one end, and they
play an important role in understanding the behavior of
flexible polymer chains at surfaces and interfaces. In
particular, the large layer thickness of the adsorbed
polymer chains is believed to be mainly governed by the
tail portions in both theoretical and experimental sides.
Moreover, the tail portions of adsorbed chains not only act
as a steric stabilization moiety for colloidal particles but
also can serve as bridges between different particles in
flocculation of colloidal dispersion^.^
Tail formation on the solid surface in adsorption experiments is available by use of (1)a polymer terminated
with a single specific functional group, which strongly
interacts with surface sites of the adsorbing surface, or (2)
a block copolymer in which one block adsorbs in train
conformation and the other block adsorbs negligibly. The
two methods described above contain some demerits: (1)
A polymer chain adsorbed with one anchored terminal
group may desorb from the surface under good solvent
conditions as the result of repulsive interaction of dangling
tails caused by the excluded volume effect if the bond
strength between the terminal group and a surface site is
not very strong. (2) One adsorbed block may not behave
as one long train part without loop formation.
However, there are a few successful examples. Hadziioannou et
recently have measured surface forces between 2-vinylpyridinestyrene block copolymers adsorbed
on mica surfaces; the 2-vinylpyridine block adsorbs
Mie University.
* Nippon Zeon Co., Ltd.
strongly on a mica surface with a flattened conformation,
while the styrene block is not adsorbed directly to a mica
surface at all. Some block copolymers were used as
polymer monolayers at the air-water interface where one
block (uncharged block) lays flat on the interface and the
other block (charged block) is extended into ~ a t e r . ~ , ~
On the other hand, there are some adsorption studies
of polymers terminated by a polar group, which adsorbs
preferentially over functional groups in the main polymer
chains."'l They used low molecular weight fractions, i.e.,
M,
2 X lo3. The effect of terminal functional groups
on adsorption was definitely observed as expected. For
high molecular weights of polymers with terminal groups,
a definite effect of terminal functional groups on adsorption behavior has not been recognized. In this study we
investigate the effect of a terminal functional group with
strong adsorbability on polymer adsorption behavior by
using well-characterized samples for both adsorbate and
adsorbent. Adsorption measurements of high molecular
weight ( lo5)polybutadienes terminated with bis(p-diethy1amino)phenyl)methanol onto the silica surface were
performed by using IR spectroscopy in carbon tetrachloride, which is a good solvent for polybutadiene, in
comparison with adsorption of normal polybutadiene. The
reason why these adsorption experiments are done under
good solvent conditions is as follows: under good solvent
conditions both the adsorbed amount and surface coverage
are usually much smaller than those under poor solvent
conditions at the same polymer concentration; thus,it is
expected that the differences in adsorbed amount as well
as surface coverage between polymers terminated and
those not terminated with functional groups will be emphasized by the preferential adsorption of terminal groups.
-
-
Experimental Section
Materials. Polybutadiene (PBR) and polybutadiene terminated with bis(p-(diethy1amino)phenyl)methanol(T-PBR)were
kindly supplied from Nippon Zeon Co., Ltd. The chemical
0024-9297/88/2221-lO59$01.50/00 1988 American Chemical Society